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Information Visualization0(0) 1–17� The Author(s) 2012Reprints and permission:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/1473871612439644ivi.sagepub.com

Recommendation and visualizationof similar movies using minimumspanning dendrograms

Michail Vlachos1 and Daniel Svonava2

AbstractExploration of graph structures is an important topic in data mining and data visualization. This work presentsa novel technique for visualizing neighbourhood and cluster relationships in graphs; we also show how thismethodology can be used within the setting of a recommendation system. Our technique works by projectingthe original object distances onto two dimensions while carefully retaining the ‘backbone’ of important dis-tances. Cluster information is also overlayed on the same projected space. A significant advantage of ourapproach is that it can accommodate both metric and non-metric distance functions. Our methodology isapplied to a visual recommender system for movies to allow easy exploration of the actor–movie bipartitegraph. The work offers intuitive movie recommendations based on a selected pivot movie and allows theinteractive discovery of related movies based on both textual and semantic features.

KeywordsData embedding, data projection, minimum spanning tree, hierarchical clustering, distance preservation,simulated annealing

Introduction

Data visualization is an inherent task in data analysis

and data mining, because it ‘forces us to see’ (in the

words of John Tukey1) hidden or obvious properties

and relationships of the data examined. The abun-

dance of highly dimensional and complex data intensi-

fies the need for advanced visualization techniques

that are able to translate the original object relation-

ships into a form easily comprehensible by the human

brain, i.e. translated into two or three dimensions.

Despite the great proliferation of visualization tech-

niques and tools, the majority of low-dimensional

visualization techniques typically focus on preserving,

in an approximate manner, all the object distances. In

that sense they end up preserving exactly none of the

original distances. The methodology presented in this

work retains a subset of important distances very accu-

rately: those belonging to the minimum spanning tree

graph. This helps preserve the local data structure and

partially also the global structure.

In addition, the method provides a novel way of

capturing both object relationships and cluster infor-

mation. Our approach fuses two-dimensional (2D)

mapping techniques with dendrograms for resolving

the cluster structure at different granularities. The

dendrogram is overlayed (either literally or metaphori-

cally) on the third dimension (i.e. extruding from the

2D page plane) to determine the cluster information.

The user can visualize the cluster information at vari-

ous granularities by projecting the cluster information

onto the 2D mapping. The user can interactively select

the cluster resolution, starting from small, low-level

clusters and gradually advancing to higher-level cluster

1IBM Zurich Research Lab, Switzerland2Slovak University of Technology, Bratislava, Slovakia

Corresponding author:Michail Vlachos, IBM Research,

structures. Finally, the proposed mapping carefully

considers both metric and non-metric distance

functions.

Besides its usefulness for general data visualization,

the proposed technique can offer meaningful ways of

data exploration in the following application areas:

– Business intelligence. An important task of

marketing teams of every large company is market

segmentation, that is, identifying groups of custom-

ers with similar buying characteristics. Our tech-

nique can simplify tasks, such as revealing

meaningful customer clusters, with the help of the

interactive multi-resolution feature it offers.

– Expanding on the above notion, the proposed data

viewing paradigm can also be fruitfully realized in the

context of recommender systems. Consider, for

example, the case of major corporations that offer a

range of products and services to their customers. It

is of great interest not only to identify groups of cus-

tomers interested in certain products but also the

‘order’ of approach to those customers (e.g., from

most likely to least likely to buy). By providing the

neighbourhood graph, tasks like the aforementioned

can be addressed in a more informative way.

This work focuses on using the proposed visualiza-

tion technique for building a movie recommenda-

tion engine on a movie–actor bipartite graph, allowing

the exploratory visualization of the graph space.

The structure of the paper is as follows. First, we

review the related work and place our work in the

proper context next to graph visualization tools and

low-dimensional embedding techniques. We continue

in the ‘Description of our approach’ section with an

overview of the proposed mapping technique. We

highlight what are the differences when metric or non-

metric distance functions are used. We also explain

how a placement technique inspired by simulated

annealing preserves the mapping quality but helps

reduce visual clutter. The ‘Application’ section illus-

trates an implementation of the proposed method

within a movie recommender system; we provide a

description of the prototype application that we have

built. We give several visual examples of the recom-

mendations and the mapping as provided by our sys-

tem. The ‘Experiments’ section evaluates the distance

preservation characteristics of our mapping. We also

provide a comparison against ISOMAP.

Related work

Graph structures are present in most scientific disci-

plines; therefore, several graph visualization techniques

have been presented over the years.2–6 Excellent sur-

veys of the various methods can be found in Refs. 7

and 8. Compared with previous work on graph visuali-

zation, the focus of this work is primarily not on scal-

ability, but on quality of representation.

Our technique presents a mapping (or projection) f

that preserves a subset of distances given a distance

matrix D, which records pairwise distances of high-

dimensional objects under a given distance function d.

The objects are originally described in a feature space

R n, but we assume we have access only to their pair-

wise distances. The distance function d can be either

metric or non-metric. Multi-dimensional scaling

(MDS)9 is one of the most well-known low-dimen-

sional mapping techniques. Objects are placed on a

lower-dimensional space so as on average the original

pairwise distances are preserved as closely as possible.

In recent years, many visualization techniques focus

on the topic of non-linear dimensionality reduction.

One of the most popular methods is ISOMAP,10 which

works in a similar manner with MDS but focuses on

preserving geodesic distances instead of the original

distances. Therefore, the minimum spanning tree

(MST) of the original distance matrix is computed

and then the geodesic distances are estimated by com-

puting the distance of each pair of objects through the

MST-participating edges. These new distances are

then projected onto a low-dimensional space using a

Principal Component Analysis (PCA)-like approach.

The above projection leads to an ‘unfolding’ of the

high-dimensional manifold, hence promoting the

visualization of complex structures on two or three

dimensions. Other similar (in spirit) projection tech-

niques include the Laplacian Eigenmaps,11 Hessian

Eigenmaps,12 the Locally Linear Embedding13 and

others.14,15 An overview of manifold projection tech-

niques has been presented by Carter.16 The goal of

those projection methods is to preserve all distances

approximately, while our approach preserves a subset

of distances (spanning-tree distances) exactly. In the

experiments we show that our methodology leads to

reduced distortion of the original distances on the

lower dimensionality.

Data-embedding techniques can also be used for

visualizing high-dimensional data. Such methods

embed arbitrary spaces into a Euclidean or pseudo-

Euclidean space. Examples include the Bourgain

embedding,17,18 FastMap19 and, recently, BoostMap.20

Locality-sensitive hashing (LSH) methods can also

be applied to achieve a low-dimensional data projec-

tion.21,22 LSH methods are generally applicable for

distance functions where locality-sensitive functions

have been identified, such as for real vector spaces

with Lp distance measures, or bit vectors when com-

pared with the Hamming distance. Therefore, LSH

2 Information Visualization 0(0)

techniques are not suitable for arbitrary distance

functions. The above shortcoming can be lifted by

employing distance-based hashing techniques.23 In

general, hash-based techniques are tailored for approx-

imate nearest-neighbour retrieval rather than low-

dimensional visualization. Moreover, only probabilistic

guarantees are given with respect to the distance pre-

servation. In comparison, our projection technique

provides deterministic guarantees for metric distance

functions, and best-effort distance preservation for non-

metric approaches.

The work presented here constitutes an expanded

version of the visualization technique presented in ref.

24. Here we explore in more detail how the proposed

high-dimensional data embedding methodology can

be used as the interface of a movie recommendation

engine. Graph-based approaches are quite prevalent

in the area of recommender systems.25,26 Several rec-

ommendation systems have appeared in the literature

in recent years, particularly for recommending

videos27,28 or movies.29–32 Visual-driven interfaces

have also been presented in the field of recommender

systems.33 Recently, researchers have been exploring

the use of data-embedding approaches, particularly in

the setting of collaborative filtering.34 In our scenario,

we use a movie–actor database as the underlying graph

structure. Given a selected pivot movie the system can

retrieve a set of similar movies that are portrayed and

clustered on two dimensions. Our approach allows the

exploratory visualization of the movie graph space and

incorporates additional multimedia capabilities, such

as automatic streaming of the selected movie trailers.

In addition, the presented prototype showcases

advanced filtering and selection functionalities. The

final outcome is an interactive movie recommendation

engine that allows the facile and meaningful explora-

tion of the movie–actor bipartite graph.

Description of our approach

The proposed method combines the visual simplicity

and comprehensibility of the MST with the grouping

properties of hierarchical clustering approaches (den-

drograms). Given pairwise distances describing the

relationship of high-dimensional objects, the objective

is to preserve a subset of important distances as well as

possible on two dimensions, while at the same time

accurately conveying the cluster information.

Our approach works as follows. First, we construct

a minimum spanning tree layout on the 2D plane in

such a way that all distances to a user-selected pivot

point and the neighbourhood distances on the tree are

exactly preserved. This construction carefully consid-

ers how to best portray the original object relationships

for either metric or non-metric distance functions.

Second, the dendrogram cluster hierarchy is con-

structed so that it can be positioned exactly on top of

the MST mapping of objects. The cluster hierarchy

can be frozen at any resolution level (tomographic

view) to convey the multi-granular clusters that are

formed. This concept is elucidated in Figure 1: objects

are properly mapped on the 2D plane whereas on the

third dimension is portrayed the hierarchy of the clus-

tering structure. Naturally, this is only a conceptual

Figure 1. Conceptual illustration of our approach. The spanning tree structure is preserved and projected onto a 2Dplane, while cluster structure is overlayed in the third dimension. ‘Cutting’ the dendrogram at a certain level projectsthe cluster structure onto 2D.

Vlachos and Svonava 3

illustration of our approach. In practice, cluster infor-

mation is also projected onto two dimensions, for

example by properly colouring the nodes belonging to

the same cluster. Therefore, by ‘cutting’ the derived

dendrogram on a user-defined level, clusters are

formed on the two-dimensional plane. Clusters are

expanded or contracted appropriately, as the user drills

up or down on the cluster hierarchy.

The advantages of the proposed technique include:

1. Preservation of both local and global structure.

2. Concurrent visualization of neighbourhood

and cluster structure.

3. Multi-granular data visualization. The user can

select the desirable detail of visualization (zoom

in/zoom out capability). This is beneficial in its

own right, when deciding, for example, the

validity of a cluster instance.35

4. Ease of interpretation. Our technique adapts on

widely utilized visualization primitives, with which

users have interacted before. From a usability per-

spective, this is beneficial for the user, as many

psychological studies clearly suggest that ‘expo-

sure to similar conditions lays the groundwork for

quicker and deeper comprehension’.36

5. Our technique is agnostic on the distance func-

tion. The input to our technique can be any

distance matrix between objects and the dis-

tance function can be either metric or non-

metric (i.e. obey or not the triangle inequality).

Neighbourhood preservation

We will first explain how to capture on two dimensions

the relationship between a set of high-dimensional

objects. Not all pairwise distances can be preserved on

two dimensions. In d dimensions one can preserve

exactly d distances with respect to a given object.10

Here, we choose to maintain, as well as possible, the

spanning tree distances, because these distances offer

valuable information about both the local relationships

and the general global structure.37,38

We begin by constructing the spanning tree on the

original high-dimensional objects. One object is

selected as the pivot and mapped in the centre of the

2D plane coordinate system. By traversing the span-

ning tree, objects are positioned on the 2D plane by

triangulating the distances to two objects: the pivot

object and the neighbouring point previously mapped

on the spanning tree.

Example. Suppose the first two points (A and B) of

the MST have already been mapped, as shown in

Figure 2(a). Let’s assume that the second distance

preserved per object is the distance to a reference

point, which in our case is the first point. The third

point is mapped at the intersection of circles centred

at the reference points. The circles are centred at A

and B with radii of d(A,C) – the distance between

points A and C – and d(B,C), respectively. Owing to

the triangle inequality, the circles either intersect in

two positions or are tangent. Any position on the inter-

section of the circles will retain the original distances

towards the two reference points. The position of

point C is shown in Figure 2(a). The fourth point is

mapped at the intersection of the circles centered at A

and C (Figure 2(b)) and the fifth point is mapped

similarly (Figure 2(c)). The process continues until all

the points of the MST are positioned on the 2D plane,

and the final result is shown in Figure 2(d).

The mapping technique presented will retain exactly

the distances between all points and the pivot

sequence, as well as between the nodes that lie at the

edges of the spanning tree. This creates a powerful

visualization technique that not only allows preserva-

tion of the nearest neighbour distances (local struc-

ture), but, in addition, retains distances with respect

to a single reference point, providing the option of a

global data view using that object as a pivot.

Figure 2. Two-dimensional embedding of the spanning tree of objects.


Algorithm 1 NodeMap(G, p) function

Require: G=(V ,E) is the original graph, (V ,Et) is

the created MST, p 2 V is the pivot object, S is a

stack

Ensure: 8v2V :Pos(v)2R2, L2(pos(v),pos(p))=Ew(v,p)Pos(p) (0, 0)//distribute children in a regular star-like shape

for all ci 2 children(Et, p) do

Pos(ci) (0,E(v, c))rotate(Pos(ci),

2p�ijchildren(p)j )

S:push(ci)end for

//process the rest of the graph in DFS order

while jSj . 0 do

t S:pop()PositionChildren(p, t, children(Et, t))for all ci 2 children(Et, t) do

S:push(ci)end for

end while

return Pos

Algorithm. We provide a pseudocode of the mapping

procedure in Algorithm 1. The input to the mapping

algorithm is the spanning tree on the given pairwise dis-

tances and the user-defined pivot object. The pivot

object is positioned in the origin of the coordinate sys-

tem. Subsequently, the algorithm positions its children in

a star-like pattern, each in its corresponding distance to

the pivot with uniform angular spacing between consecu-

tive children. The remainder of the graph is processed in

depth–first–search (DFS) order. Once a node is popped

from the stack, all its children are positioned by triangu-

lating the already mapped positions of the pivot and

parent node. The PositionChildrenfunction

(Algorithm 2) demonstrates the triangulation process. Itcalls the CircleCircleIntersection function, which returnsthe intersection points between two circles. Note that thisversion of the function refers to the case when a metricdistance function is used and the circles are guaranteed tointersect. To handle the non-metric case, we introduce theproper extensions in subsequent sections.

The two circles will intersect in at most two points.

The current pseudocode from the two candidate inter-

sections selects the one with the lower azimuth value

with respect to the currently expanded node. This is

indicated by the Left position and defined inAlgorithm 3. Once a node position is mapped on 2D,all its children nodes are pushed onto the stack, andthe process repeats until no nodes remain in the stack.

Algorithm 2 PositionChildren(p, t,C)

Require: G=(V ,E), Pos(p),Pos(t) 2 R2, C � V

// Pos(p) are the root (pivot) coordinates and Pos(t)are the parent coordinates

// C is a set of children of the node t

Ensure: 8v 2 C : Pos(v) 2 R2

// Pos() contains the 2D coordinates for all children

nodes

for all ci 2 C do

//find candidate points that preserve the distances

E from the root and pivot nodes

//by computing the intersection of the two dis-

tance circles

I CircleIntersect(Pos(p),E(p, ci),Pos(t),E(t, ci))//position the current child at the left candidate

Pos(ci) Left(I , t)end for

Algorithm 3 Left/ Right(P , t)functions

Require: 8e 2 P : e 2 R2, Pos(t) 2 R2

Ensure: point e0 2 R2 minimizes the azimuth of the

vector e0 � Pos(t) for the Left variation of this func-

tion and maximizes it for the Right variation

return e0 : min=maxfAzimuth(e0 � Pos(t)) : 8e0 2 Pg

Layout optimization using simulated annealing

The focus of the previously presented mapping

approach was the preservation of distances on the

spanning tree. However, when dealing with data visua-

lization, other issues need to be addressed as well. One

such issue is clarity of display,39 in an effort to minimize

the cognitive load.40

For achieving maximum visual clarity, it is advanta-

geous to minimize the number of intersecting graph

edges. Recall that when triangulating the position of a

third point to its neighbour and the pivot, the algorithm

proceeds by identifying the intersection between two

circles. One can readily see this in Figure 3; unless the

two circles are tangent, there are two positions in which

a newly mapped point can be placed (we cover the

non-metric case where circles do not later intersect).

The core node mapping algorithm as presented in

Algorithm 1 uses the PositionChildren function, which

greedily picks the intersection based on the azimuth.

In order to select more intelligently which of the two

mapping positions to follow, we adapt an optimization

process that leads in reduction of the number of edge

intersections. We employ a probabilistic global optimi-

zation technique based on simulated annealing (SA)

principles.41 SA is an effective optimization method

when the search space is discrete, which is the case in

our scenario of interest.

SA algorithms utilize ‘heating’ and ‘cooling’ primi-

tives to avoid the situation of the objective function

getting stuck at a local minimum. Heat causes random

moves to states of higher energies, whereas a slower

cooling provides chances for convergence to configura-

tions with lower energy.

In each step the current solution found by the SA

algorithm is randomly transformed to a nearby


solution in the state space. The magnitude of change

is random but proportional to the temperature t. The

newly generated candidate solution is scored by the

objective function. If it performs better than the cur-

rent solution, it is kept as the reference solution for the

next iteration. Owing to the vast size of the state space

in our setting, we do not permit the state expansion in

a direction that does not improve the score.

In our setting, we need to decide which of the two

positions of the intersecting circles to choose. Therefore,

the decision state space is a binary vector of length N

indicating which of the two intersection points (denoted

as Left or Right) to choose for all N edges of the span-ning tree. The current state in the decision space is storedas a vector CS. The function PositionChildren thatcalculates the node placement dictated by a given binaryvector CS is outlined in Algorithm 4.

Algorithm 4 PositionChildren SA(P , t,C,CS)

Require: G=(V ,E), Pos(p),Pos(t) 2 R2, C � V ,

8e 2 E : CS(e)=TRUEjFALSE

// CS is a binary vector defining the mapping for a

given tree

//keeping one bit of information for every Left/Right

placement decision

Ensure: 8v 2 C : Pos(v) 2 R2

// Pos()contains the 2D coordinates for all children

nodes

for all ci 2 C do

I CircleIntersect(Pos(p),E(p, ci),Pos(t),E(t, ci))if CS(E(t, ci))=TRUE then//query the control

bit for a given edge

Pos(ci) Left(I, t)else

Pos(ci) Right(I, t)end if

end for

The function CircleIntersect produces thecandidate intersection points for every mapped childnode. The node is placed at the Left (least azimuth)intersection point if the value of the bit of CS that rep-resents the decision for the current edge equals TRUE.Otherwise the node is positioned at the Right intersec-tion (larger azimuth position).

Algorithm 5 SALayout function

Require: G=(V ,E), p 2 V

Ensure: 8v 2 V : Pos(v) 2 R2// Pos(v)contains the

2D coordinates for all nodes in V

8e 2 E : CS(e) TRUE

T 1//initial temperature

e desired numerical accuracy

while T . e do

CSnew CS

for all e 2 E do

if Random½0, 1�\ T then

CSnew(e) :CS(e)end if

end for

Pos=NodeMap SA (CS)Posnew =NodeMap SA (CSnew)if EdgeCrossings(Posnew)\ EdgeCrossings(Pos)then

//layout defined by CSnew contains less edge

crossings, move the search to CSnew

CS CSnew

T T �aelse

T T �aend if

end while

Pos=NodeMap SA (CS)

SA is performed on the control vector CS, direct-

ing the bit state position towards values that minimize

the number of intersected edges of the spanning tree. A

pseudocode for this process is given in Algorithm 5.

During each iteration, the modified state vector CSnew

is evaluated by the layout algorithm and directed

towards the state that minimizes the number of edge

intersection. First, the node layouts Pos and Posnew, dic-

tated by CS and CSnew respectively, are calculated using

the function NodeMap_SA. To eliminate repetition wedo not rewrite this function. It is identical with the onedescribed in Algorithm 1, with the exception that

Figure 3. Selecting which of the two positions a new pointis mapped to can affect the number of subsequentintersections with other edges . As the triangulationindicates two candidate positions for each point that needsto be mapped, there exist a large number of potentiallayouts that satisfy the triangulation mapping process.


PositionChildren is replaced withPositionChildren_SA, which is its bit-vector driven coun-terpart as provided previously in Algorithm 4. Finally,function EdgeCrossings counts the number of inter-secting edge pairs for a particular layout. If a candidatelayout results in a lower number of intersected edges, it iskept for the next iteration. The temperature is increased(T T �a) so that additional opportunities can beexplored and local minima potentially avoided.Conversely, if the candidate solution does not lead to abetter solution, the global temperature is reduced(T T=b), with the cooling-off indicating that we maybe close to reaching the global minimum.

In the experimental section, we show that the simu-

lated annealing process significantly reduces the num-

ber of intersected edges on the 2D projected space.

Triangulation on non-metric distances

So far, it was assumed that the underlying distance

measure obeys the triangle inequality; therefore, the

circles around the reference points are guaranteed to

intersect. However, many widely used distance func-

tions (e.g. dynamic time warping,42 longest common

subsequence43) violate the triangle inequality, and thus

the corresponding reference circles may not necessarily

intersect. In such cases, one needs to identify the posi-

tion of where an object should be placed with respect

to the two circles, in such a way that the object is

mapped as close as possible to the circumference of

both circles. We need to identify the locus of points that

minimize the sum of distances to the perimeters of two

circles. One can show that the desired locus always lies

on the line connecting the centres of the two circles.

We highlight necessary extensions that allow its

proper usage under non-metric distances. We can

identify two cases for the non-intersecting circles:

– Case 1: the two circles are disjoint and one does

not enclose the other; and

– Case 2: one circle encloses the other.

One can show the following:

Lemma 1 The point that minimizes the sum of distances

to the perimeters of two circles lies on the line L connecting

the two circle centres and is located midway on the line seg-

ment with vertices at the intersection of L with the circles’

perimeters.

Case 1 is shown in Figure 4. The midpoint C is

given by the equation:

C =A1 +A2 � A1

d� d + r1 + c � r2

2ð1Þ

where c= � 1, A1 is the circle with larger radius,

d = jA2 � A1j and r1, r2 are the radii of the corre-

sponding circles.

For case 2, where one circle encloses the other, one

can identify two subcases:

– When the two circles have disjoint centres, as

shown in Figure 5(a), then the coordinates of the

sought point C that minimizes the distance to the

two circles is also given by Equation (1), where in

this case c =1.

– When the two circles have common centres,

then there exist two points per line that satisfy

the distance minimization property, as shown in

Figure 5(b). There is an infinite number of such

lines, all passing through the common centre of

the two circles.

C=A1+cos(a) �sin (a)sin(a) cos(a)

24

35 � (0, r1+ r2

2), a2 0,2p½ �

The above modifications allow the positioning of the

objects on the 2D plane, so that our mapping approach

also accommodates cases where a non-metric distance

function has been used to construct the input distance

matrix.

The derived formulae are combined to implement

the function CircleCircleIntersection_NM

in Algorithm 6. This function can replace insidePositionChildren the metric case ofCircleCircleIntersection algorithm, when theutilized distance measure is non-metric.

Figure 4. Circles that are disjoint.


Algorithm 6 CircleIntersect NM(c1:c, c1:r, c2:c, c2:r)function

Require: c1:c 2 R2, c1:r 2 R, c1:c 2 R

2, c2:r 2 R

Ensure: (x, y) 2 R2

e desired numerical accuracy

D L2(c1:c, c2:c)if D \ e then

//circles are concentric

return (0, c1:r + c2:r2

)else if D . c1:r + c2:r then

//circles c1 and c2 do not intersect, nor overlap

return c1:c+ c2:c� c1:c=D�c1:r � c2:r +D=2

else if D \ jc1:r � c2:rj then

//circles c1 and c2 do not intersect, but one is con-

tained in the other

return c1:c+ c2:c� c1:c=D�c1:r + c2:r +D=2

else

//triangle inequality holds, use regular circle

intersection

return CircleIntersect(c1:c, c1:r, c2:c, c2:r)end if

Cluster preservation

We now turn our attention on how to embed cluster

information on the already mapped MST graph.

Recall that the input for the algorithm is a matrix of

pairwise distances. Based on the given pairwise dis-

tances one can build a hierarchical dendrogram. The

dendrogram construction is based on a single linkage

approach that merges closest singleton objects and clus-

ters. The process iterates as follows:

1. The two closest items objects are merged together

into one cluster and these merged objects are deleted.

2. The distances of the remaining objects to the

newly formed cluster are recomputed. The dis-

tance is selected to be the minimum distance of

an object to any of the cluster objects. This

step is important because it allows us to exactly

overlay the resulting dendrogram.

3. The process is repeated until only one cluster

object remains.

The hierarchical cluster construction that we adapt

updates the distances in such a way so that minimum

distances between objects are favoured in the merging

process. This allows us to effectively combine the two

techniques. One can show the following lemma:

Lemma 2 The merging order of the minimum spanning

tree algorithm is the same as the merging order of the single

linkage hierarchical clustering approach.

What the above essentially tells us is that we can

achieve a perfect overlay of the constructed hierarchy

on top of the MST-mapped points, as the clustering

order is the same as the order crystallized in the

spanning-tree mapping. This fact can be also seen in

Figure 1 where the single linkage dendrogram is posi-

tioned exactly on top of the spanning-tree mapping.

The above observation, combined with the hierarchi-

cal cluster information, provides the user with the ability

to have multi-granular cluster view on two dimensions,

by interactively imposing variable thresholds on the

resulting dendrogram. This allows the creation of a

‘tomographic view’ of the clusters, and promotes the

interactive formation of incrementally larger or smaller

cluster structures. The concept is illustrated in Figure 6.

By cutting the dendrogram using a smaller threshold,

six clusters are created on the left. Imposing

Figure 5. One circle encloses the other.


progressively higher thresholds leads to merging of the

clusters, as shown on the right side of the same figure.

Clusters can be conveyed on two dimensions either

enclosed in ellipses or by forming the convex hull of

the contained objects; another approach is through the

means of colour. In our prototype implementation we

convey cluster information by properly colouring the

node perimeter and also the connected edges.

Overall complexity. Naively, the proposed algorithm

comprises two stages: the hierarchical clustering and

the creation of the spanning tree. When the single-

linkage clustering (SLC) and MST algorithms are

used, these stages can be performed separately and as

the algorithms are inherently similar and thus produce

compatible output, the resulting dendrogram can be

positioned over the 2D-mapped spanning tree.

Upon closer inspection of the clustering algorithm,

one can realize that the edges that initiate the cluster

merging in the SLC algorithm are equivalent to the

spanning tree produced by MST. This implies that

there is no need to perform the actual MST algorithm

on the original graph. The runtime complexity of the

layout algorithm presented in Algorithm 1, dominated

by the depth-first traversal of the spanning tree, is

O(n). The resulting complexity of our approach is

therefore dominated by the clustering algorithm itself,

which, in the case of SLC, is O(n2 log n) where the

O( log n) component is due to the operation of the

Disjoint-Set data structure. The complexity of the lat-

ter component can be reduced to O(a(n)), where a is a

certain inverse of Ackermann’s function, as shown by

Chazelle.44 Many graphs with high node cardinality,

where the O(n2) component could impose a problem,

are in practice sparse (e.g. social networks) where there

can be established tighter bounds on the number of

edges as a constant multiple of graph node count, for

example O(e)=O(k�n)=O(n). Then, the overall run-

ning complexity is O(na(n)). Finally, Karger et al.45

presented a randomized MST algorithm that runs in

linear time with high probability using only edge-

weight comparisons.

The mapping and clustering algorithm can be com-

bined with the simulated annealing layout optimiza-

tion (‘Layout optimization using simulated annealing’

section) to improve the visual clarity. The process of

clustering and spanning tree discovery is separate from

the layout algorithm. Therefore, the resulting com-

plexity is O(Oorig + n�I) where Oorig is the original

complexity discussed above and I is the number of

iterations of the layout algorithm required by the opti-

mization method to converge. In the ‘Experiments’

section we demonstrate that for graphs with approxi-

mately 100 nodes it is reasonable to expect low cardin-

ality of iterations in the order of 100 – 200 for the SA

portion of the code.

Figure 6. Cluster information conveyed by variable thresholds on a dendrogram information, and thus imposing ‘zoomin’ and ‘zoom out’ in the cluster structure.


In practical applications the proposed visualization

technique is not applied on an arbitrary number of

graph nodes, because this would pose a screen flood-

ing problem. The minimum spanning dendrogram is

typically constructed as the outcome of a k-nearest-

neighbour (k-NN) search with respect to the exam-

ined pivot. For k-NN operations an index structure

(B-tree or R-tree) is typically employed, allowing for

the expedient retrieval of the k-neighbourhood with

respect to the pivot point. Therefore, the final visuali-

zation is ultimately constructed on top of a few hun-

dred objects. This allows the very effective runtime

operation of our solution.

Application: visualizing a movie–actorgraph

Using the proposed methodology we have built a

movie recommendation engine that visualizes a graph

of related movies based on the pivot movie selected.

We use a movie graph comprising approximately

125,000 movies and 950,000 actors. Attributes of the

movies include features such as title, year, genre, direc-

tor, plot, language and cast, to name only a few. Each

movie is enriched with additional textual features

extracted from the internet. Therefore, each movie

record is represented by a set containing keywords/

phrases pertaining to the movie.

To evaluate the similarity between two movies, we

consider how similar are the textual sets describing

these movies. We compute the distance d using the

Jaccard coefficient J for the textual sets of movies x

and y:

d = 1� J(x, y)= 1� jx [ yjjx \ yj

The above distance is a metric distance func-

tion.46,47 Other functions for evaluating the similarity,

such as cosine similarity48 or Latent-Semantic-

Analysis (LSA)49 can also be directly accommodated.

However, the above simple approach worked very effi-

ciently in practice and provided intuitive results, while

having the added benefits of efficient execution. In

addition, the Jaccard metric can work well even when

comparing objects with different numbers of attri-

butes, such as in our scenario (variable set of textual

features per movie).

Graphical user interface and functionalities. The data

visualization is accommodated through a graphical

web interface (Figure 7), which incorporates the

neighbourhood and cluster visualization technique

described in the preceding sections. The interface

allows the user to easily search for a movie with the aid

of an autocomplete interface. It subsequently searches

for the k-NN neighbours of the selected movie, where

the parameter k is user selectable. For the set of neigh-

bours the proposed visualization graph is created, pla-

cing the movie selected as the centre (pivot) object.

The user can then easily identify other relevant movies,

with the option to retrieve detailed information about

the movie, such as participating actors or the movie

plot. The user can easily modify the number of dis-

played movie clusters or even watch the movie trailer

which is automatically retrieved from the Internet. The

application allows the exploration of both sides of the

Figure 7. The web-based graphical user interface of the movie recommender system application.


movie–actor bipartite graph: either by deeper explora-

tion of the movie graph (by clicking on a movie) or by

searching/filtering for movies of a particular actor (by

clicking on an actor’s image).

Additional filtering functionalities include:

– filtering by rating, for example when interested in

retrieving movies with a rating . X; and

– filtering by year, when the user is interested only

in recent movie releases.

Below we provide specific examples of our map-

ping, highlighting its ability to be used as an effective

and interactive movie recommendation system.

Clusters are conveyed using varying border and edge

colours. In practice, obviously movies could be illu-

strated by their DVD covers.

Examples. Selecting The Matrix as the pivot movie,

our paradigm will correctly pack The Matrix 2 and The

Matrix 3 to it (Figure 8). An adjacent cluster contains

the ‘Animatrix’, an anime movie inspired by the

Matrix trilogy. We can also identify a cluster of movies

belonging to The Terminator series of movies. Other

relevant movies that can be found on the graph include

Minority Report, Strange Days, Total Recall and Blade,

among others.

Placing Titanic as the pivot movie produces the 2D

mapping shown at the top of Figure 9. The user can

navigate through the graph and identify similar movies.

Clustered with Titanic are the movies Titanic (1953),

Poseidon and Shakespeare in Love. Another cluster dis-

played, shown in light green, includes movies such as

The Notebook, Atonement and Purple Rain; all romantic

drama movies.

Similarly, selecting Star Wars as the pivot movie cor-

rectly packs closely the remaining Star Wars movies

(Figure 9, bottom). An adjacent cluster includes paro-

dies of Star Wars, such as Spaceballs or Thumb Wars.

Other related movies shown on the graph include adven-

ture films such as The Lord of the Rings trilogy or science-

fiction action thrillers such as Aliens and Star Trek.

Experiments

We evaluate how the SA layout algorithm reduces the

number of edge intersections. We also highlight the

ability of our algorithm to preserve the original object

distances and provide a thorough comparison with the

mapping provided by the popular ISOMAP

algorithm.10

SA layout algorithm

We assess the quality of the node layout placement

with and without the proposed SA layout algorithm.

Recall that the SA algorithm uses an optimization

function to properly place the mapped nodes, so as to

minimize the number of intersected edges (Figure 10).

We use the dataset from the ‘Application’ section,

which represents each movie with a bag of words.

From this set we randomly select 1000 objects. For

Figure 8. The Matrix selected as pivot object for our mapping.


each we retrieve the k-NNs for k equal to 20, 30, 40

and 50. The distance is computed using the Jaccard

coefficient as described in the ‘Application’ section.

We consider each of the objects/movies as pivots and

use the proposed method to visualize the relationship

of the retrieved neighbourhoods, with and without the

Figure 9. Using the proposed mapping on a movie graph. Top: The pivot movie is Titanic. Bottom: Star Wars is selectedas the pivot movie.


SA layout placement. The SA algorithm significantly

reduces the number of intersected edges. This leads to

better readability of the graph of mapped objects.

Naturally, the SA algorithm incurs additional computa-

tional expense, because it introduces an additional

number of iterations. The additional time required by

the SA process is given in Table 1 (note that the SA

component is executed within the web browser, and

therefore the reader should emphasize the relative times

for different k values rather than the absolute times).

The table also reports the median number of edge

intersections and the number of iterations for the layout

algorithm, with and without the annealing component.

Figure 10. Examples of how the simulated annealing component improves the visual layout and reduces the number ofedge intersections.

Table 1. Node placement using simulated annealing (SA)significantly reduces the number of edge intersections(median number shown). We also report the additionaltime incurred by the SA process.

k-value Edge intersections SA iterations SA time

WithoutSA

WithSA

(median[min–max])

(sec)

20 36 12 102 [0–248] 0.4

30 108 48 146 [59–335] 1.4

40 222 112 146 [59–335] 4.3

50 346 212 146 [59–349] 4.4


We provide the median, minimum and maximum num-

ber of iterations. The median number of iterations

ranges between 100 and 150 iterations, indicating a

low complexity for the SA component. We note that

both variations provide the same distance preservation

with respect to the pivot and the MST distances.

However, the SA implementation clearly enhances the

visualization capacity of the mapping by providing a

more intelligent node placement.

Distance preservation in projected space

The mapping presented in this work has been designed

to accurately preserve on two dimensions the following

two distance types:

– the distance of all points to the pivot (center)

object; and

– the distance of all neighbouring points on the MST.

In addition to the above cases, we also evaluate the

distortion of the distance for all pairs of objects. This

is a criterion that our approach does not directly opti-

mize. However, because we preserve several important

distances exactly, we expect that the remaining ones

do not deviate significantly.

We also put forward a comparison with the 2D map-

ping provided by ISOMAP. ISOMAP represents a

method for performing a non-linear mapping of

D-dimensional points, on a lower dimensionality d. It

has attracted significant interest from the data-mining

and machine-learning community because of its effec-

tiveness and simplicity of implementation. The mapping

performed by ISOMAP attempts to ‘unfold’ the data,

when they lie on a high-dimensional manifold. It shares

some commonalities with the approach presented in this

work, because ISOMAP achieves the data unfolding by

computing spanning-tree distances in the original space,

and preserving them on the d-dimensional space

through a PCA-like projection operation.

However, the process of preserving on average all

spanning tree distances eventually is the weak point of

ISOMAP. The final projection, even though revealing

about the overall data structure, eventually results in

significant distortions of the original distances. Here,

we show that while our mapping preserves perfectly

the pivot and neighbouring MST distances, ISOMAP

results in a projection of lower quality.

Experiment. From our movie database we pick ran-

domly 1000 movies as pivot objects and then retrieve

the k = 20,50,100 nearest neighbours, as suggested by

the distance function. For each object, we evaluate how

close the projected distances are on two dimensions,

compared with the original high-dimensional ones. We

identify three subsets of distances: (a) distances of all

objects to the pivot, (b) distances of neighbouring

spanning-tree distances and (c) all pairwise distances.

We make the above distinction because our approach

carefully preserves the first two types of distances.

ISOMAP requires one parameter as the input: the neigh-

bourhood for creating the local graph of each object.

Selecting this parameter is not an easy task, because it is

data dependent. Typically, low values that lead to a fully

connected graph are preferable. For each experiment

we set the neighbourhood value to k=3, because lower

values led to disconnected graph components.

Our findings are reported in Figure 11. Distance

preservation is calculated as the mean ratio of dis-

tances in the projected space over distances in the

original space. That is, if M is the original matrix of

pairwise distances and M are the distances on the 2D

space, then we estimate the ratio r =mean(M=M).Values of r closer to 1 indicate that the projected dis-

tance on two dimensions is closer to the original dis-

tance. The ratios r for the corresponding 1000 graphs

are sorted from the lowest to the highest values.

The experiments indicate that distances to the pivot

object are accurately preserved; the distance preserva-

tion reaches 100% (Figure 11, top left). Distances on

the MST graph are also perfectly retained, as indicated

on the top-right graph of Figure 11. This is not sur-

prising, because our method performs the mapping in

such a way, so that pivot and MST distances are pre-

served. In contrast, ISOMAP results in significantly

lower performance regarding preservation of original

MST distances.

Finally, we evaluate what is the accuracy of our tech-

nique for preserving all pairwise object distances. Our

methodology does not directly minimize the distortion

of all pairwise distances on 2D. However, by preserving

the spanning tree distances, our methodology essen-

tially preserves the ‘backbone’ of the dataset structure.

Therefore, the remaining distances do not generally

deviate significantly from the original distances.

This result is conveyed on the bottom-left graph of

Figure 11. As expected, both our mapping and

ISOMAP do not perfectly preserve pairwise distances.

Because from this graph it is not apparent which

approach provides a better mapping, we present the

same results on a win–lose graph (bottom-right side).

For each of the 1000 experiments this graph indicates

when a mapping is closer to the original distance. For

k = 20, our approach provides a better mapping than

ISOMAP for 45% of the cases. For larger neighbour-

hoods this situation reverses. Our methodology is bet-

ter for 75% and 87% of the experiments for k = 50

and k = 100, respectively. This final result implies that

for large graph instances ISOMAP mapping provides

a lower quality of mapping, because as the algorithm


attempts to preserve all distances it eventually distorts

the original graph structure.

In conclusion, these experiments attest to the abil-

ity of our mapping technique to accurately preserve

on low dimensions the original high-dimensional

graph structure. We have verified empirically the

high quality of mapping, in particular compared with

other popular graph projection techniques, such as

ISOMAP.

Current limitations and future work

The proposed mapping algorithm does not preserve

all pairwise distances, but only distances to the central

(pivot) object, as well as the neighbouring distances on

the MST. Therefore, points that are distant on the

MST graph have been mapped with respect to differ-

ent neighbouring points. No guarantees can be made

for the distance of these points on the 2D mapping.

See, for example, in Figure 8 the movies Star Wars:

Episode 3 and Live Free or Die Hard. A tentative solu-

tion for this could be the following: as mapped points

get more distant from the pivot, the algorithm can

position the objects by preserving the weighted dis-

tance towards multiple points. This approach, of

course, leads again to approximate distance preserva-

tion. Another related shortcoming of the technique is

that even though it promotes exact distance preserva-

tion, this may hinder visualization, for example at

highly clustered areas. A way to mitigate this could be

Figure 11. Comparison with ISOMAP with respect to distance preservation in the projected space. Top left: Ourapproach exactly preserves all distances to the pivot object. Top right: Spanning distances are also very accuratelypreserved. Bottom left: Pairwise distances are generally better preserved by ISOMAP. However, for smallneighbourhoods (k = 20), our approach results in an embedding that is only 2% worse than ISOMAP. Bottom right:win–lose graph for all-pair distance preservation.


to relax the constraint of exact distance preservation.

Then, the projected points of the spanning tree would

be adjusted in order to promote better filling of the

space. By considering a histogram of the distances of

all objects toward the pivot object, this could be

addressed as a histogram equalization process.

Conclusion

This work presented a new visualization paradigm for

graphs and high-dimensional data, in general. Our

approach combines both neighbourhood preservation

and cluster preservation. Neighbourhood distances are

virtually identical to the original high-dimensional dis-

tances with respect to a designated pivot point. We put

forward several node layout policies, including a prob-

abilistic SA placement algorithm, which retains the

same distance preservation properties, while minimiz-

ing the number of intersected edges in the graph struc-

ture. This significantly enhances the visualization power

of the lower-dimensional projected mapping. Finally,

we have shown that our approach can effectively sup-

port both metric and non-metric distance functions.

A unique characteristic of our approach is that it

exploits the commonalities between spanning-trees

and single linkage hierarchical dendrograms, thus

being able to capture on the projected graph both

neighbourhood and cluster structure. In this manner

our technique allows the interactive growth and

shrinking of clusters at progressively granular levels.

This facilitates the effective zooming in and out of the

identified data clusters.

We have shown that the proposed technique offers

superior mapping capabilities in comparison with pre-

viously used visualization techniques, such as

ISOMAP. We have applied our visualization approach

on a movie–actor graph, and built an interactive movie

recommendation engine that showcases the above

principles.

Funding

The research presented here was partially supported

by a Marie Curie Reintegration Grant.

Acknowledgement

This work was conducted while the second author was

visiting the IBM Zurich Research Lab.

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